An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the usual orthogonality relation is an orthogonality space. We present simple conditions to characterise the orthogonality spaces that arise in this way from finite-dimensional Hermitian spaces. Moreover, we investigate the consequences of the hypothesis that an orthogonality space allows gradual transitions between any pair of its elements. More precisely, given elements e and f, we require a homomorphism from a divisible subgroup of the circle group to the automorphism group of the orthogonality space to exist such that one of the automorphisms maps e to f, and any of the automorphisms leaves the elements orthogonal to e and f fixed. We show that our hypothesis leads us to positive definite quadratic spaces. By adding a certain simplicity condition, we furthermore find that the field of scalars is Archimedean and hence a subfield of the reals.

正交空间是具有对称和自反二元关系的集合。任何配备自反和各向异性内积的线性空间都提供了一个例子：一维子空间的集合和通常的正交关系是一个正交空间。我们提出了简单的条件来表征以这种方式从有限维厄米空间中产生的正交空间。此外，我们研究了正交空间允许其任何一对元素之间逐渐过渡的假设的后果。更准确地说，给定元素e和f，我们需要从圆群的可分子群到正交空间的自同构群的同态存在，使得自同构之一映射e到f，并且任何自同构都使与e和f正交的元素固定。我们证明我们的假设将我们引向正定二次空间。通过添加某个简单条件，我们进一步发现标量场是阿基米德的，因此是实数的子场。